Blender plays a short .mp3 sound when it completes the render - HOW?
The aud module

For a standard  the ubiquitous wave module to play ".wav" files would be a failsafe way to go. 

However, Blender comes with the aud module which plays any format of sound that can be used in blender. My build supports:

&gt;&gt;&gt; bpy.path.extensions_audio
frozenset({'.aac', '.oga', '.mka', '.eac3', '.aiff', '.aif', '.wav', '.mp2', '.ogg', '.flac', '.ac3', '.mp3', '.m4a', '.wma'})


Test script, edit the path to match path to audio file.

import bpy
import aud

path = "/home/batfinger/Music/CCR Chronicle, Vol. 1 - The 20 Greatest Hits/DISC/Creedence Clearwater Revival - Suzy Q. (Part 1) - 13.mp3"
# load the sound
snd = aud.Sound(path)
dev = aud.Device()

def finish_render(scene):
    dev.play(snd)
    return None

# clear others, assuming this is only one.    
bpy.app.handlers.render_complete.clear()
# assign handler.  
bpy.app.handlers.render_complete.append(finish_render)


Here is the way to stop the playback.

&gt;&gt;&gt; import aud
&gt;&gt;&gt; dev = aud.Device()
&gt;&gt;&gt; dev.stopAll()